Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent

Wang, Hongbin. "Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent." Czechoslovak Mathematical Journal 66.1 (2016): 251-269. <http://eudml.org/doc/276798>.

@article{Wang2016,
abstract = {Let $\Omega \in L^s(\{\mathrm \{S\}\}^\{n-1\})$ for $s\ge 1$ be a homogeneous function of degree zero and $b$ a BMO function. The commutator generated by the Marcinkiewicz integral $\mu _\Omega $ and $b$ is defined by \begin\{equation*\} \displaystyle [b,\mu \_\Omega ] (f)(x)=\biggl (\int ^\infty \_0\biggl |\int \_\{|x-y|\le t\} \frac\{\Omega (x-y)\}\{|x-y|^\{n-1\}\}[b(x)-b(y)]f(y) \{\rm d\} y\bigg |^2\frac\{\{\rm d\} t\}\{t^3\}\bigg )^\{1/2\}. \end\{equation*\} In this paper, the author proves the $(L^\{p(\cdot )\}(\mathbb \{R\}^\{n\}),L^\{p(\cdot )\}(\mathbb \{R\}^\{n\}))$-boundedness of the Marcinkiewicz integral operator $\mu _\Omega $ and its commutator $[b,\mu _\Omega ]$ when $p(\cdot )$ satisfies some conditions. Moreover, the author obtains the corresponding result about $\mu _\Omega $ and $[b,\mu _\Omega ]$ on Herz spaces with variable exponent.},
author = {Wang, Hongbin},
journal = {Czechoslovak Mathematical Journal},
keywords = {Herz space; variable exponent; commutator; Marcinkiewicz integral},
language = {eng},
number = {1},
pages = {251-269},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent},
url = {http://eudml.org/doc/276798},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Wang, Hongbin
TI - Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 1
SP - 251
EP - 269
AB - Let $\Omega \in L^s({\mathrm {S}}^{n-1})$ for $s\ge 1$ be a homogeneous function of degree zero and $b$ a BMO function. The commutator generated by the Marcinkiewicz integral $\mu _\Omega $ and $b$ is defined by \begin{equation*} \displaystyle [b,\mu _\Omega ] (f)(x)=\biggl (\int ^\infty _0\biggl |\int _{|x-y|\le t} \frac{\Omega (x-y)}{|x-y|^{n-1}}[b(x)-b(y)]f(y) {\rm d} y\bigg |^2\frac{{\rm d} t}{t^3}\bigg )^{1/2}. \end{equation*} In this paper, the author proves the $(L^{p(\cdot )}(\mathbb {R}^{n}),L^{p(\cdot )}(\mathbb {R}^{n}))$-boundedness of the Marcinkiewicz integral operator $\mu _\Omega $ and its commutator $[b,\mu _\Omega ]$ when $p(\cdot )$ satisfies some conditions. Moreover, the author obtains the corresponding result about $\mu _\Omega $ and $[b,\mu _\Omega ]$ on Herz spaces with variable exponent.
LA - eng
KW - Herz space; variable exponent; commutator; Marcinkiewicz integral
UR - http://eudml.org/doc/276798
ER -